<p>In this paper we prove that any full Perazzo algebra <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1555_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$A_F$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>F</mi> </msub> </math></EquationSource> </InlineEquation>, whose Macaulay dual generator is a Perazzo form <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1555_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="233" /> </InlineMediaObject> <EquationSource Format="TEX">$F\in K[X_0,\dots ,X_n,U_1,\dots ,U_m]_d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>∈</mo> <mi>K</mi> <msub> <mrow> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mn>0</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>U</mi> <mi>m</mi> </msub> <mo stretchy="false">]</mo> </mrow> <mi>d</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1555_Article_IEq3.gif" Format="GIF" Height="43" Rendition="HTML" Resolution="72" Type="Linedraw" Width="169" /> </InlineMediaObject> <EquationSource Format="TEX">$n+1 = \left( {\begin{array}{c}d+m-2\\ m-1\end{array}}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mfenced close=")" open="("> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>d</mi> <mo>+</mo> <mi>m</mi> <mo>-</mo> <mn>2</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, is the doubling of a 0-dimensional scheme in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1555_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {P}^{n+m}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mi>m</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> and we compute the graded Betti numbers of a minimal free resolution of <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10231_2025_1555_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$A_F$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>F</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

Betti numbers of full Perazzo algebras

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2025年2月26日

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R. M. Miró-Roig,
Josep Pérez

摘要

In this paper we prove that any full Perazzo algebra $A_F$ $A_{F}$ , whose Macaulay dual generator is a Perazzo form $F\in K[X_0,\dots ,X_n,U_1,\dots ,U_m]_d$ $F \in K {[X_{0}, \dots, X_{n}, U_{1}, \dots, U_{m}]}_{d}$ with $n+1 = \left( {\begin{array}{c}d+m-2\\ m-1\end{array}}\right) $ $n + 1 = (\begin{matrix} d + m - 2 \\ m - 1 \end{matrix})$ , is the doubling of a 0-dimensional scheme in $\mathbb {P}^{n+m}$ $P^{n + m}$ and we compute the graded Betti numbers of a minimal free resolution of $A_F$ $A_{F}$ .